Viral kinetics, stability and sensitivity analysis of the within-host COVID-19 model

This paper delves into the investigation of the COVID-19 dynamics within a host using the Target-Latent-Infected-Virus (TLIV) model, presenting a fresh approach compared to previous studies. Our model introduces a latent class and explores sensitivity analysis, aspects that have received limited attention in prior research. A significant contribution of this study is the analysis of both local and global stability of equilibrium states, subject to specific sufficient conditions based on the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}R0. By examining these stability properties, we aim to gain insights into the factors underlying variations observed in the findings of different studies. Additionally, we identify the death rate of infected cells as the parameter most susceptible to influence in our model. To minimize its impact and facilitate recovery, it is crucial to implement appropriate medical therapies and consume immune-boosting foods. Some computer simulations are carried out to strengthen the theoretical results.

Within-host COVID-19 model. Based on current data, it is estimated that around 80% of  cases are classified as mild to moderate, and individuals in these cases typically experience a full recovery from the infection 24 . Previous research has indicated that the humoral response, involving the production of antibodies, to SARS-CoV-2 infection is commonly observed in individuals who have been infected. The level of antibodies, specifically the anti-SARS-CoV-2 IgG titers, has been strongly linked to the extent of virus-specific CD4+ and CD8+ T cell responses in the bloodstream 25 . However, it should be noted that most samples of convalescent plasma, collected from individuals recovering from COVID-19, did not exhibit high levels of neutralizing activity. Furthermore, only a small number of individuals who were analyzed showed the presence of rare antibodies that possess potent antiviral activity against specific viral proteins 26 .
The angiotensin-converting enzyme 2 (ACE2) receptors present on the surface epithelial cells are bound by the SARS-CoV-2 virus. These cells, which express ACE2, are considered susceptible to viral infection and are referred to as target cells in mathematical models. The distribution of these target cells varies notably across different sections of the respiratory system, with the lungs having the highest density, followed by the nose, and finally, the tissues of the trachea/bronchi. Consequently, COVID-19 patients often experience pneumonia, making it a relatively frequent occurrence 27,28 .
Our proposed model in this paper comprises four variable quantities, the susceptible targeted cells, T(t), latent cells, L(t), infected cells, I(t), and free virus particles, V(t). The model assumes that there is a constant of regeneration d 1 T(0) susceptible targeted cells. These cells are infected by free virus particles with a bilinear incidence rate βT(t)V (t) and these infected cells produce with a rate p free virus particles. Parameters d 1 , d 2 , d 3 , and d 4 are the death rate of susceptible target cells, latent cells, infected cells, and free virus particles, respectively. Latent cells on average span 1/k units of time in L class and then join the infected class of cells. It should be noted that d 1 is a natural death rate or natural clearance rate while d 2 , d 3 and d 4 are a combination of the natural clearance rate and the role of the immune system in the elimination of these cells.
Our work presents a novel mathematical model that incorporates latent class analysis to describe the dynamics of COVID-19 within the host. Unlike previous studies that focused solely on estimating the basic reproductive number 29 , or analyzing the stability of equilibrium states 30 , our model delves deeper into the impact of various parameters on the system. By conducting sensitivity analysis, we have identified the death rate of infected cells as the most sensitive parameter in our model. This finding sheds light on the critical role this parameter plays in shaping the dynamics of COVID-19 infection and highlights its potential implications for disease progression and management strategies.
Authors in 29,30 have studied the viral kinetics of COVID-19 without latent class of cells, we consider the mathematical within-host model in the form The originality of our model lies in the inclusion of a new class, namely the latent cells, to capture the dynamics of COVID-19 within the human body. Within this class, individuals or cells have been exposed to the disease but have not yet reached the infectious state. The duration of this latent period, determined by the delay parameter, signifies the time it takes for individuals or cells to become infectious. This approach is applicable to more general fractional systems, see 31 .
In the traditional model, only the susceptible, infected, and viral load compartments were considered. However, by introducing the latent class, we aim to provide a more realistic representation of the disease progression. This addition accounts for the incubation period during which the virus replicates within the body before symptoms become apparent. By incorporating the latent class, our model captures an essential aspect of the disease phenomenon, enhancing its accuracy and predictive capabilities. It can serve as a valuable tool for studying the effectiveness of interventions and informing public health strategies aimed at controlling the spread of  Here are the model assumptions: 1. In COVID-19, the respiratory system's cell population remains constant due to the slow turnover of respiratory epithelial cells. 2. d 1 T(0) is assumed to be a constant of regeneration of cells. 3. Once cells recover from the infection and become immune, the model assumes that they remain immune indefinitely. 4. Cell variations in susceptibility are not considered. 5. During the latent period, cells are considered non-infectious or have a low probability of transmitting the disease to other cells.
(1) www.nature.com/scientificreports/ 6. There are no major changes in the characteristics or behavior of cells within the latent class during the modeling period. Graph of the system. By performing parameter estimation in a similar way to 29,30 , using the Monte Carlo Markov Chain (MCMC) method, and in light of the chest radiograph score data, the estimated values of parameters are T(0) = 50, β = 1, d 1 = 0.2, d 2 = 2, d 3 = 1.1, d 4 = 3, k = 1.7 and p = 0.05 . In this scenario, the graph of our model (1) can be sketched in Fig. 1 in 30 days. Basic properties of the system like the nonnegativity and uniform boundedness of the solution are shown in the next section. The equilibrium states of the model and the basic reproduction number are provided in Section "Equilibrium states and the basic reproduction number". Local and global stability of the equilibrium states E 0 and E * with some computer simulations are shown in Section "Extinction and persistence". Viral kinetics of the COVID-19 model using sensitivity and elasticity analysis of R 0 towards the effective parameters in the model are shown in Section "Sensitivity and elasticity of R 0 ". Conclusions and some future works are devoted to Section "Conclusion and further directions.

Nonnegativity and uniform boundedness of the solutions of (1)
In this section, we study the nonnegativity and the uniform boundedness of the solutions of the within-host model (1) with respect to the initial conditions The next theorem proving nonnegativity and uniform boundedness in the mathematical model of cell population dynamics ensures valid interpretations and prevents unrealistic predictions, supporting reliable analysis and decision-making in cell biology and related fields. (2), then all solutions of (1) are nonnegative and uniformly bounded.

Proof
The first equation of model (1) is  Regarding the boundedness of the solutions of (1), the total population of cells and Consequently, all solutions of (1) with respect to (2) are bounded in a biologically feasible region Clearly, the number of free virus particles is also bounded at any time t.

Equilibrium states and the basic reproduction number
Equilibrium states of this model satisfy the following algebraic equations Clearly, we have two equilibrium states at most, the infection-free equilibrium state . The basic reproduction number R 0 is defined as the spectral radius of F 0 V −1 0 , consequently Then for R 0 > 1 , there is a unique positive equilibrium that can be written again in the form

Extinction and persistence
In this section, we shall perform the stability analysis, in terms of the basic reproduction number R 0 , we study the extinction and the persistence of the disease inside the human body by focusing on the local and global stability of E 0 and E * . Evaluating the Jacobian matrix J at the equilibrium states, the LaSalle-invariance principle and the Routh-Hurwitz criterion 33 are used for investigating the local stability of these equilibrium states. Introducing appropriate Lyapunov functions around the equilibrium states is very helpful in investigating the global stability of these states 34 .
Extinction. Theorem 4.1 If R 0 < 1 , the infection-free equilibrium E 0 is locally asymptotically stable. And if Proof The Jacobian matrix for the right hand side of (1) is Evaluating J at the infection-free equilibrium E 0 Let i be the eigenvalues of this matrix for i = 1, 2, 3, 4 such that  www.nature.com/scientificreports/ where Then dV dt < 0 if R 0 < 1, and dV dt = 0 if only V = 0 or R 0 = 1 . According to the LaSalle-invariance principle 34 , the equilibrium state E 0 is globally asymptotically stable.
The theorem's implications are significant for disease control strategies, as it offers insights into conditions favoring infection eradication and emphasizes the significance of minimizing the basic reproduction number ( R 0 ) to prevent outbreaks and ensure the well-being of the cell population.
The Euler method scheme is utilized for simulating the within-host COVID-19 model 35    www.nature.com/scientificreports/ the other compartments are shown by the phase planes in Fig. 2b-d. The disease dies out within the human body for R 0 < 1.

Persistence. Theorem 4.2 The positive endemic equilibrium E * is locally asymptotically stable and globally
asymptotically stable if R 0 > 1.

Proof Evaluating J at the positive endemic equilibrium implies
The characteristic equation is Now, for R 0 > 1, and Then according to the Routh-Hurwitz criterion, the positive endemic equilibrium E * is locally asymptotically stable. Regarding the global stability of E * , choose the Lyapunov function Then Using the equations  www.nature.com/scientificreports/ As the arithmetic mean is greater than or equal to the geometric mean, then V < 0 and V = 0 only if S = S * , L = L * , I = I * and V = V * . Then according to the LaSalle-invariance principle, the positive endemic equilibrium E * is globally asymptotically stable.
Using the same initial values of the numerical simulation in Fig. 2a, and

Sensitivity and elasticity of R 0
The basic reproduction number R 0 quantifies the average number of secondary cases generated by the introduction of a disease into a susceptible population. It is a vital measure for assessing the potential for disease transmission 39 . Many papers have used optimal control strategies and sensitivity analysis of many disease models including COVID-19, see 40,41 . In our study, we examine the impact of the basic reproduction number R 0 on the dynamics of infected cells and free virus particles over a span of six weeks, as illustrated in Fig. 4. During this time interval, we observe that higher values of R 0 lead to a greater peak in the number of infected cells and free virus particles. This suggests that a higher R 0 is associated with a more pronounced spread of the infection within the host. Furthermore, in Fig. 5, we present surface plots that depict the relationship between R 0 and the various parameters incorporated into our model. These plots allow us to visualize how changes in these parameters www.nature.com/scientificreports/ influence the value of R 0 , providing insights into the factors that impact the transmissibility and progression of the infection.
In order to gain insights into the factors that have the most significant impact on the number of infected cells and free virus particles, we conducted a sensitivity analysis. This analysis allowed us to determine the sensitivity of the basic reproduction number R 0 with respect to a specific parameter ω by calculating the derivative ∂R 0 ∂ω . By plotting the dynamics of infected cells I(t) and free virus particles V(t) for various values of the infection rate β , we observed an interesting trend: when the infection rate was low, the number of infected cells and free virus particles exhibited a slower rate of decrease. Additionally, we found that the sensitivity of parameter d 2 and the latent period 1 k have a similar impact to β , which is demonstrated in Fig. 6.    www.nature.com/scientificreports/ of infected cells d 3 . As we mentioned previously, this death rate represents a combination of natural clearance mechanisms and the role of the immune system. Therefore, it becomes essential to consider treatments that can reduce the infection within the host. Notably, increasing the death rate of infected cells leads to a decrease in the number of free virus particles, as depicted in Fig. 6. To further examine the influence of d 3 , the most sensitive parameter, we plot the variation of R 0 with different values of other parameters in Fig. 7. This analysis provides valuable insights into the interplay between key parameters and their impact on the spread of the infection within the host. Another control measure is the elasticity index which measures the change of R 0 with respect to the change in the parameters. The elasticity of R 0 with respect to the parameter ω can be calculated by ϒ ω Consequently, The elasticity of R 0 exhibits positive correlations with respect to the parameters β, p and k. Decreasing these parameters leads to a reduction in the value of R 0 , indicating that the infection can be effectively eliminated from the human body. Conversely, negative correlations are observed with parameters d 2 and d 3 . This implies that increasing the death rates of latent cells and infected cells, possibly through treatments such as plasma therapy, monoclonal antibodies, and immune system-boosting foods, can contribute to the eradication of the disease. These correlation relationships, highlighting the impact of parameter changes on R 0 , are visualized in Fig. 8. Understanding these relationships is crucial for designing effective interventions and treatment strategies to combat the infection.
There is a positive correlation between R 0 and T(0) which means a larger initial number of target cells T(0) in a limited cell population can lead to a higher probability of infection and potentially faster virus spread.
Based on this discussion, there are additional real applications to the considered problem. Firstly, in Treatment Optimization, the sensitivity analysis identifying the death rate of infected cells as a critical parameter can inform the development of enhanced treatment strategies, leading to improved patient outcomes. Secondly, in Vaccine Development, the incorporation of latent class analysis into our model enables its use in assessing and refining COVID-19 vaccine candidates, aiding in the development and evaluation of effective vaccination strategies. Lastly, in Risk Assessment and Management, our model's sensitivity analysis offers valuable insights into the relative significance of various parameters in shaping disease outcomes. This information can assist in conducting risk assessments, informing mitigation strategies, and efficiently allocating healthcare resources.

Conclusion and further directions
In this study, we present a novel formulation of a within-host COVID-19 mathematical model. Unlike previous studies, our model takes into account the latent class, providing a new perspective on disease dynamics. We examine the nonnegativity and ultimate boundedness of the analytical solution to gain insights into the behavior of the system. Furthermore, we investigate the extinction and persistence of the disease within the human body by analyzing the local and global stability of equilibrium states, specifically E 0 and E * . To support our findings, we www.nature.com/scientificreports/ perform numerical simulations and validate the results through the examination of sensitivity and elasticity of R 0 concerning the model's parameters. Our results reveal that the disease dies out when R 0 < 1 without treatment, whereas it persists when R 0 > 1 . Of notable importance is the death rate of infected cells, which emerges as a highly sensitive parameter that can be enhanced through appropriate medical therapy and a diet that supports the immune system. Additionally, we propose that our research can be expanded to incorporate discrete and distributed delays, further enhancing our understanding of the disease dynamics. Moreover, fractional models offer a more accurate representation of complex and heterogeneous systems. They capture the intricate nature of disease transmission, taking into account factors such as varying susceptibility, heterogeneous mixing patterns, and non-integer order derivatives 42,43 .

Data availability
All data used in this manuscript have been presented within the article.

Code availability
The codes used and/or analysed during the current study available from the corresponding author on reasonable request.
Received: 1 December 2022; Accepted: 13 July 2023 Figure 8. Elasticity of R 0 allows to assess the sensitivity of the basic reproduction number to variations in specific parameters.